We have explained the anomalous temperature dependence observed in the electron spin relaxation rates of three low spin hemoproteins. The data indicate a temperature dependence of T6.30 plus or minus 0.07. Our explanation assumes that the protein vibrations which couple most strongly to the Fe3 ion result from the motion of the polypeptide chain. Neglecting the variation in amino acid residues, the protein is a polymer of N identical monomers. The fractal dimension d, the spatial extent R of the protein, and N are related by N infinity Rd, where d depends upon how the basic units of the protein arrange themselves in space. Even when d is non-integer, we have shown that a T3 plus 2d dependence is expected. In ordinary crystalline solids d equals 3 and a T9 temperature dependence is observed. From our data we deduce a fractal dimension of 1.65 plus or minus 0.04 for the proteins. Within experimental error this number is the decimal equivalent of 5/3 which is the computed fractal dimension associated with a self-avoiding random walk on a three dimensional lattice. An analysis of existing x-ray of the positions of the alpha carbons on the protein backbone, confirms our relaxation result by yielding d equals 1.67 plus or minus 0.04. Additional experiments confirming our model are in progress.